Search Results (158)

In this paper, we introduce a novel class of Fubini–Fibonacci–Appell polynomials within the framework of Golden calculus. Utilizing generating function techniques and fibonomial convolution methods, we establish several structural properties, including explicit series representations, summation formulas, convolution identities, and recurrence relations involving the Golden derivative. Furthermore, we construct three-dimensional extensions and derive determinant representations for these polynomials. As special cases, we identify connections with Bernoulli–Fibonacci, Euler–Fibonacci, and Genocchi–Fibonacci polynomials. The results presented herein unify and extend several known Fibonacci-type polynomial families and provide a systematic Appell-type approach in Golden calculus. Full article

In this paper, we introduce and investigate a new class of r-circulant matrices whose entries are generated by higher-order Fibonacci numbers. Explicit representations of the eigenvalues of these matrices are derived by means of the Binet formula together with the structural properties of r-circulant matrices. Based on these representations, a closed-form expression for the determinant is obtained. In addition, several summation identities involving higher-order Fibonacci numbers are established, including formulas for partial sums, sums of squares, and weighted sums. These identities play a fundamental role in the derivation of the norm expressions and spectral estimates of the matrices. Furthermore, several matrix norms, including the Euclidean (Frobenius) norm, the 1-norm, the ∞-norm, and the spectral norm, are investigated in detail. Lower and upper bounds for the spectral norm are obtained for both cases $\left|r\right|\ge 1$ and $\left|r\right|<1$ by employing Hadamard product techniques and classical norm inequalities. Finally, numerical examples are presented to illustrate and validate the theoretical results. Full article

In this paper, we introduce and investigate a new class of special polynomials called degenerate Peter–Genocchi polynomials. We define these polynomials and their associated numbers via an explicit generating function and explore a variety of their fundamental algebraic and analytic properties. In particular, we derive summation formulas (expressing polynomials via their associated numbers), addition formulas (splitting the polynomial argument), an implicit summation formula (a bivariate convolution identity), and symmetric identities. We also establish connections with degenerate Stirling numbers of both kinds, higher-order degenerate Genocchi polynomials, and higher-order degenerate Daehee polynomials. Furthermore, we investigate derivative properties and present a differential operator formula. Finally, we provide tables of approximate zeros and graphical representations of the zeros in the complex plane. Full article

This paper introduces a new family of extended degenerate Bell-based Appell polynomials by applying Euler’s integral as a fractional operator to the Appell-type degenerate Bell polynomials. Beginning with the exponential operational rule and employing the fractional operator framework, this paper derives the operational connection, generating function, explicit summation formula, determinant representation, complete four-step proof via Cramer’s rule, recurrence relations, and the monomiality principle with raising and lowering operators and rigorously verifies the commutation relation. Applications to the extended degenerate Bell–Bernoulli, Bell–Euler, and Bell–Genocchi polynomials are presented as three numbered examples, each with generating functions; operational connections; determinant representations; numerical first values; and graphical illustrations including surface plots, zero diagrams, and stacked zero figures. Full article

This article introduces and develops a comprehensive theory of the Mixed Degenerate Gould–Hopper–Appell Type Polynomials $MDGHA$-$TPs$, constructed by embedding an Appell factor into the framework of degenerate Gould–Hopper generating functions. Beginning with the generating function formulation, we derive explicit series representations, monomial-type operational identities, recurrence relations, and a determinantal form that encodes the algebraic structure of the family. Summation identities expressed via Stirling numbers of the first kind and addition-type formulas are established. A detailed numerical investigation of the zero distributions of these polynomials is then carried out, with graphical illustrations revealing symmetry patterns and geometric arrangements in the complex plane. Connections with classical sequences of Appell, Hermite, and Gould–Hopper are explored throughout. The article concludes with remarks on open problems including the orthogonality of the $MDGHA$-$TPs$ with respect to suitable weight functions, the asymptotic behaviour of their zeros as the degree tends to infinity, and potential applications to boundary-value problems in heat diffusion, perturbation expansions in quantum mechanics, and signal processing in non-homogeneous media. Full article

This study presents the development of a comprehensive framework wherein probabilistic and analytical techniques collaboratively produce identities pertinent to special functions. The fundamental premise is that ratios of random variables, particularly within the family of Gamma and Beta distributions, inherently generate integral representations and hypergeometric structures that can be harnessed to derive non-trivial summation formulas. A pivotal outcome of this investigation is that, in the context of independent and identically distributed random variables, symmetry plays a crucial role. A key property of ratios of i.i.d. random variables translates into straightforward probabilistic assertions that directly lead to precise analytic identities, thereby enabling the recovery of classical results such as Kummer’s and Watson’s summation theorems as consequences of distributional symmetry. When the assumption of identical distribution is relaxed, hypergeometric representations continue to exist; however, the absence of symmetry impedes closed-form evaluations of the same nature. This contrast underscores symmetry as the fundamental mechanism underlying exact summation formulas and elucidates why such identities are contingent upon specific parameter configurations. More generally, this methodology offers a probabilistic interpretation of hypergeometric identities and establishes a conceptual connection between probability theory and the theory of special functions. Furthermore, it suggests that more expansive constructions, based on non-identically distributed variables or iterated processes, could be fruitfully explored. In particular, examining ratios may lead to new identities or alternative derivations of classical results. Full article

In this paper, a new class of special polynomials, called the Sheffer-type general-$\lambda$-matrix polynomials, is introduced within the framework of the monomiality principle. This family is obtained by combining the structure of Sheffer sequences with the theory of general-$\lambda$ matrix polynomials, which leads to a unified formulation encompassing several polynomial families. Fundamental properties of the proposed polynomials are established, including their generating function, explicit series representation, summation formulas, quasi-monomial structure, differential relations, and determinant representation. The proposed framework addresses an important problem in the theory of special functions: the systematic construction of matrix-valued polynomial families that simultaneously generalize both classical scalar polynomials and existing matrix polynomial hierarchies. Such a unified structure is of broad significance, with applications in quantum mechanics (wave function expansions), mathematical physics (matrix differential equations and spectral problems), approximation theory, and the study of special functions in the matrix domain. Several hybrid forms of the proposed family are derived through appropriate choices of the defining functions, which yield polynomial subclasses related to classical families such as Hermite, Laguerre, Bessel, and Poisson–Charlier polynomials. These subclasses illustrate how the proposed framework provides a systematic approach for constructing and studying generalized polynomial structures. In each case, the matrix parameter L introduces a new layer of structural richness not present in the scalar setting, enabling the modelling of phenomena governed by matrix-valued spectral data. Furthermore, a numerical and graphical investigation of selected hybrid forms is carried out using Mathematica (version 14.3, 2025; Wolfram Research, Inc.). Surface plots, distributions of complex zeros, and real-zero patterns are presented for different parameter values, highlighting the influence of the parameters on the behavior and structural characteristics of the polynomials. Full article

This study uses a fractional operator technique to analyze a novel class of special polynomials. These polynomials are designated as fractional Gould–Hopper–Bell–Apostol-type polynomials. We first define the operational expression of the Apostol-type Gould–Hopper–Bell polynomials and then use a suitable fractional operator to generate a new fractional version of these polynomials. The accompanying generating function, series definition, and summation formulas are also derived. Furthermore, certain symmetry identities and monomiality results are investigated. The study also identifies specific members of this fractional family, such as fractional Gould–Hopper–Bell–Apostol–Bernoulli polynomials, fractional Gould–Hopper–Bell–Apostol–Euler polynomials, and fractional Gould–Hopper–Bell–Apostol–Genocchi polynomials, and finds similar results for each. The study makes use of Mathematica to display computational results, zero distributions, and graphical demonstrations for a specific case of the established class. Full article

In recent years, special function theory has played an increasingly important role in the development of advanced mathematical models and statistical distributions. In this paper, a new extension of the Euler Beta function is introduced by employing the Wright function as a kernel, leading to the formulation of the Beta–Wright function. Several fundamental properties of the proposed function are systematically investigated, including summation formulas, functional relations, Mellin transforms, integral representations, and derivative formulas. Furthermore, extended forms of Gauss and confluent hypergeometric functions are constructed within this framework. In addition to its theoretical significance, the proposed function is applied to statistical modeling, and the associated distributions are analyzed using graphical and analytical techniques. The obtained results demonstrate that the Beta–Wright function provides a flexible and effective tool for both analytical investigations and statistical applications. Full article

This paper introduces a novel multidimensional half-discrete Hardy–Hilbert-type inequality that simultaneously addresses several key extensions in the literature. The inequality incorporates a general parameterized kernel involving a scalar term and the $\beta$-norm of a vector, and replaces the traditional discrete coefficient with a partial sum. Under suitable parameter conditions, the resulting inequality is sharper and preserves the optimal constant factor. The proof employs a systematic combination of weight-function techniques, parameter introduction, real-analysis methods, and the Euler–Maclaurin summation formula. Equivalent characterizations of the best possible constant are provided, and several meaningful corollaries are deduced, thereby unifying and generalizing a series of earlier inequalities. Full article

We systematically exploit a new generalized hypergeometric identity to obtain new hypergeometric summation formulas. As a consistency test, alternative proofs for some special cases are also provided. As a byproduct, new summation formulas with finite sums involving the psi function and a recursive formula for Bateman’s G function are derived. Finally, all the results have been numerically checked with MATHEMATICA. Full article

We examine a useful hypergeometric transformation formula by means of the coefficient extraction method. A large class of “binomial/harmonic series” (of convergence ratio “$1/4$”) containing the cubic central binomial coefficients and harmonic numbers is systematically investigated. Numerous closed summation formulae are established, including a remarkable series about harmonic numbers of the third order. Full article

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle). Full article

The probability density function (PDF) of the product of two normal random variables remains an open discussion. Researchers have proposed many forms of PDFs. Among these, two notable PDFs are an analytical solution with infinite summation and an integral form with transformation. For practical computation, they must be estimated. The form with infinite summation must be truncated to a finite summation, and the form still in integration must be computed numerically. As a result of this estimation, an error occurs in the value of the estimation. This paper derives upper bounds for the estimation error resulting from truncation and numerical approximation in integral calculations. The upper bound error between the exact PDF and the truncated PDF is expressed as a geometric series using Bessel function inequality and Stirling’s approximation. The geometric formula allows the quantification of the total truncation error to be determined. For the PDF, which is still in integration form, the trapezoidal rule is used for numeric calculation. Hence, the error can be determined using the error-bound formula. The two estimated PDFs have their own advantages and disadvantages. The truncated PDF gives a relatively small upper bound value compared to the numerical calculation integral form PDF for a small value domain. However, the truncated PDF fails to perform for a large value domain, and only the integral form PDF can be used. The error for the estimation is applied to the conventional mass measurement. The results demonstrate that the error can be controlled through an analytical approach. Full article

Cloud detection is an indispensable step in satellite remote sensing of cloud properties and objects under the influence of cloud occlusion. Nevertheless, interfering targets such as snow and haze pollution are easily misjudged as clouds for most of the current algorithms. Hence, a robust cloud detection algorithm is urgently needed, especially for regions with high latitudes or severe air pollution. This paper demonstrated that the passive satellite detector Advanced Geosynchronous Radiation Imager (AGRI) onboard the FY-4A satellite has a great possibility to misjudge the dense aerosols in haze pollution as clouds during the daytime, and constructed an algorithm based on the spectral information of the AGRI’s 14 bands with a concise and high-speed calculation. This study adjusted the previously proposed cloud mask rectification algorithm of Moderate-Resolution Imaging Spectroradiometer (MODIS), rectified the MODIS cloud detection result, and used it as the accurate cloud mask data. The algorithm was constructed based on adjusted Fisher discrimination analysis (AFDA) and spectral spatial variability (SSV) methods over four different underlying surfaces (land, desert, snow, and water) and two seasons (summer and winter). This algorithm divides the identification into two steps to screen the confident cloud clusters and broken clouds, which are not easy to recognize, respectively. In the first step, channels with obvious differences in cloudy and cloud-free areas were selected, and AFDA was utilized to build a weighted sum formula across the normalized spectral data of the selected bands. This step transforms the traditional dynamic-threshold test on multiple bands into a simple test of the calculated summation value. In the second step, SSV was used to capture the broken clouds by calculating the standard deviation (STD) of spectra in every 3 × 3-pixel window to quantify the spectral homogeneity within a small scale. To assess the algorithm’s spatial and temporal generalizability, two evaluations were conducted: one examining four key regions and another assessing three different moments on a certain day in East China. The results showed that the algorithm has an excellent accuracy across four different underlying surfaces, insusceptible to the main interferences such as haze and snow, and shows a strong detection capability for broken clouds. This algorithm enables widespread application to different regions and times of day, with a low calculation complexity, indicating that a new method satisfying the requirements of fast and robust cloud detection can be achieved. Full article